Geometry Notes 1.4, 1.5, 1.6, 1.7 PDF

Notes 1.4: Segments and their measure Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths.  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. The points on a line can b...

Notes 1.4: Segments and their measure Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths.  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. The points on a line can be paired, one-to-one, with a real number. The real number that corresponds to a point is called the coordinate of the point. The above allows us to calculate distance. The distance between points A and B is the absolute value of the difference of their coordinates. We use the notation AB for distance or the length of segment AB. AB =| x2 − x1 | Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. Find MP.  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. congruent segments— The symbol ≅ means congruent.  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. We mark congruent segments in a figure with exactly the same number of tick marks. Use the figure to write the congruent segments and the equal distances.  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. If BC = 2 feet, find AD. BC = AD, so if BC = 2 feet, then AD = ______. If DC = 6 feet, find AB. DC = AB, so if DC = 6 feet, then AB = _______.  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. Given the figure, is segment BG ≅ segment AC?  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. Postulate: Segment Addition Postulate If point B is between points A and C, then AB + BC = AC. Also, if AB + BC = AC, then point B is between points A and C. part + part = whole  Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. The midpoint of a segment is a point that divides, or bisects, a segment into two congruent segments. True statements: B is the midpoint of AC. Line m bisects AC. !""# BD bisects AC. !!!" DB bisects AC. AB ≅ BC. A line, ray, segment, or plane AB ≅ BC. that intersects a segment at its midpoint is called a segment bisector. Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. If MP = 47 units, find MN and NP. Targets: Understand the Measure of Segments and Use Segment Postulates and Algebra to Find Segment Lengths. Point C is the midpoint of segment AB. Find AC, CB, and AB.  Notes 1.5—Angles and their measure Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate  Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Definition of Angle An angle consists of two different ______________ with a common _______________ Sides of an angle— sketch an angle and label the parts vertex of an angle— Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Different ways to name this angle: Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate interior of an angle— exterior of an angle— Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate How many different angles are in the diagram? Write two other ways to name ∠1. Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Protractor The instrument shown is called a protractor. It can be used to measure angles in units called degrees (°). For example, the measure of ∠A (also denoted by m∠A) is 30°. Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Protractor Postulate  Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate right angle acute angle obtuse angle straight angle  Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Find m∠RQM, m∠RQS, and m∠RQN. Then classify each angle as acute, right, obtuse, or straight.  Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate congruent angles—  Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Use the figures to answer each question. a. If m∠C = 45°, find m∠D. b. If m∠3 = 113°, find m∠4. Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Angle Addition Postulate If P is in the interior of ∠ABC, then  Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate Example A circular pizza can easily be cut into 8 slices because of the ease of using a straight knife, as shown below. Find the angle measure of a slice of pizza cut into 8 equal-size slices, as shown. Learning Target: Find measures of angles using Algebra and the Angle Addition Postulate If ∠DEG is a right angle, find m∠DEF and m∠FEG. Notes 1.6: Angle pairs Learning target: apply algebra techniques to classify angle relationships Learning target: apply algebra techniques to classify angle relationships adjacent angles— vertex— Learning target: apply algebra techniques to classify angle relationships vertical angles— Learning target: apply algebra techniques to classify angle relationships linear pair—  Learning target: apply algebra techniques to classify angle relationships complimentary angles—  Learning target: apply algebra techniques to classify angle relationships supplementary—  Learning target: apply algebra techniques to classify angle relationships Use the figure to answer each statement as true or false. a. ∠2 and ∠3 are vertical angles. b. ∠2 and ∠4 are vertical angles  Learning target: apply algebra techniques to classify angle relationships Use the figure to answer each statement as true or false. c. ∠3 and ∠4 form a linear pair. d. ∠3 and ∠1 form a linear pair.  Learning target: apply algebra techniques to classify angle relationships Use the figure to identify each pair of complementary angles.  Learning target: apply algebra techniques to classify angle relationships Use the figure to identify each pair of supplementary angles. Learning target: apply algebra techniques to classify angle relationships Given that m∠P = 73°: a. If ∠A and ∠P are supplementary angles, find m∠A. b. If ∠B and ∠P are complementary angles, find m∠B.  Learning target: apply algebra techniques to classify angle relationships angle bisector—  Learning target: apply algebra techniques to classify angle relationships Use the figure shown to find the measure of each unknown angle. a. Find m∠BAC. b. Find m∠BAE.  Learning target: apply algebra techniques to classify angle relationships In the figure, ray QY bisects ∠RQZ. Find the value of x; then find m∠RQY and m∠YQZ.  Learning target: apply algebra techniques to classify angle relationships Solve for x and y. Then find the measure of each angle.  Notes 1.7: Midpoint and distance formulas Learning target: apply the midpoint and distance formulas to calculate geometric measures  Learning target: apply the midpoint and distance formulas to calculate geometric measures From previous sections in this chapter, we know the definition of the term “midpoint”. Write your definition below. Check with the person next to you.  Learning target: apply the midpoint and distance formulas to calculate geometric measures Midpoint Formulas On a Number Line The coordinate of the midpoint is the average or mean of the coordinates of the endpoints. Formula Given on a number line: The coordinate of the midpoint M of is  Learning target: apply the midpoint and distance formulas to calculate geometric measures On the Coordinate Plane The coordinates of the midpoint are the average of the x-coordinates and the average of the y-coordinates of the endpoints. Learning target: apply the midpoint and distance formulas to calculate geometric measures Use the diagram and find the coordinate of the midpoint, M, of segment PQ. Learning target: apply the midpoint and distance formulas to calculate geometric measures Find the midpoint of the line segment PQ, that joins points P(–3, 3) and Q(1, 0).  Learning target: apply the midpoint and distance formulas to calculate geometric measures The midpoint of segment CD is M(–2, 1). One endpoint is C(–5, 7). What are the coordinates of the other endpoint, D?  Learning target: apply the midpoint and distance formulas to calculate geometric measures The distance between two points A(x1, y1) and B(x2, y2) is The Distance Formula is based on the Pythagorean Theorem, which we will study later in this book. When we use the Distance Formula, we are really finding the length of the hypotenuse of a right triangle. Learning target: apply the midpoint and distance formulas to calculate geometric measures  Learning target: apply the midpoint and distance formulas to calculate geometric measures Find the distance between A(2, –5) and B(1, –4). Give an exact distance and a one-decimal-place approximation. 1.4 Extended Notes 𝐿 1. In the figure, line 𝑡 bisects 𝐽𝐿̅ at point K. If 𝐽𝐾 = 2𝑥 + 2, 𝐾𝐿 = 3𝑦 + 3, and 𝐽𝐿 = 4𝑥 + 𝑦 + 1, what are the values of 𝑥 and 𝑦? What are the lengths of ̅̅̅ ̅̅̅̅, and 𝐽𝐿 𝐽𝐾 , 𝐾𝐿 ̅? 𝑡 𝐾 𝐽 ̅̅̅̅. If 2. In the figure, 𝑆 is the midpoint of 𝑅𝑇 𝑅 2 𝑅𝑆 = 2𝑥 and 𝑆𝑇 = 7𝑥 + 4, then what is the value of 𝑥? What are the lengths of 𝑅𝑆 ̅̅̅̅, 𝑆𝑇 ̅̅̅̅, ̅̅̅̅? and 𝑅𝑇 𝑆 𝑇 1.5 Extended Notes 1. In the figure, 𝑚∠𝐽𝑂𝐿 = 2𝑥 2 − 𝑥 + 15 and 𝑚∠𝐿𝑂𝐾 = 5𝑥 + 5, what is the value of 𝑥? 𝐿 𝐽 𝑂 𝐾 𝐴 ⃑⃑⃑⃑⃑ bisects ∠𝐴𝐵𝐶. If 𝑚∠𝐴𝐵𝑋 = 6𝑥 + 2, 2. In the figure, 𝐵𝑋 𝑋 𝑚∠𝑋𝐵𝐶 = 5𝑦 + 2 and 𝑚∠𝐴𝐵𝐶 = 8𝑥 + 4𝑦, then what are the values of 𝑥 and 𝑦? 𝐵 𝐶 1.6 Extended Notes 1. The measure of the supplement of an is 14 less than three times the measure of the complement. Find the measure of the , its compliment, and its supplement. 2. Find 𝑥 and 𝑦 in the following figure. 3x 39 2 x 3 y 68 2 y 60 1.7 Extended Notes ̅̅̅̅. 1. Find the coordinates of the missing point given that M is the midpoint of 𝐴𝐵 7 12 𝐴 (√40, 8) , 𝑀(√160, 8 ) 2. The length of 𝐴𝐵 = √29, where pt. A has the coordinates (4, −1) and the coordinates of pt. B are (−1, 𝑘). Find 𝑘. (**HINT** There are 2 possible values for 𝑘) 1 Find the area of the shaded region. Assume all angles are right angles. Round your answers to the nearest tenth. 9 cm 3 cm 12 cm 8 cm 4 cm 2 cm 8 cm 9 cm 19 cm 2. The area of the rectangle below is 30 ft 2 , find x. 2x – 1 x+3 3. Find the EXACT perimeter of the triangle with the following coordinates of the vertices. J 3,5 , K 2,5 , L 2, 3 4. Find the perimeter of the following rectangle below. 3x 4 2x 2 y 1 2y 3